What is the integral of (sin(x))/(cos^8(x)) ?

The integral of (sin(x))/(cos^8(x)) is (sec^7(x))/7+C

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integral of (sin(x))/(cos^8(x))

\int \frac{sin ( x )}{cos ^{8} ( x )} d x

\frac{sec ^{7} ( x )}{7} + C

Solution steps

\int \frac{sin ( x )}{cos ^{8} ( x )} d x

Rewrite using trig identities

= \int \frac{tan ( x )}{cos ^{7} ( x )} d x

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{7} (x) tan (x) d x

Apply u-substitution

: \int u^{6} d u

= \int u^{6} d u

Apply the Power Rule

: \frac{u ^{7}}{7}

= \frac{u ^{7}}{7}

Substitute back

u = sec (x)

= \frac{sec ^{7} ( x )}{7}

Add a constant to the solution

= \frac{sec ^{7} ( x )}{7} + C

\int_{0}^{\frac{3 π}{2}} 4 ∣ sin (x) ∣ d x

f (x) = x^{2} e^{- 3 x}

\frac{d}{d x} (x + 7 x - 3)

3 y^{^{'}} + y = 0, y^{^{'}} (0) = 0

\frac{( e ^{2 x} - 1 )}{x}

What is the integral of (sin(x))/(cos^8(x)) ?
The integral of (sin(x))/(cos^8(x)) is (sec^7(x))/7+C